Relativistic Group Velocity Calculation

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Homework Help Overview

The discussion revolves around comparing the phase and group velocities of an electron and a proton that have the same kinetic energy (KE). Participants are exploring the implications of relativistic effects on these velocities.

Discussion Character

  • Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants are attempting to derive relationships between the velocities using relativistic equations and discussing the implications of different mass and energy terms. There are questions about the effectiveness of certain approaches and the necessity of using specific variables.

Discussion Status

The discussion is ongoing, with various participants offering different perspectives on how to approach the problem. Some have suggested alternative methods and highlighted the importance of the shared kinetic energy condition. There is no explicit consensus yet, but several lines of reasoning are being explored.

Contextual Notes

Participants are working under the assumption that the kinetic energies of the electron and proton are equal, which is central to their comparisons. There are also mentions of specific formulas and terms that may not be universally understood, indicating a potential gap in shared knowledge.

neelakash
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Homework Statement



we are given that an electron and a proton have the same KE.We are to compare their phase and group velocity...


Homework Equations





The Attempt at a Solution



K=(γ₁-1)m₁c²=(γ₂-1)m₂c²
Now, I found it very problematic to extract the ratio of v₁/v₂ in terms of m₁ and m₂
So,I expanded the γ s binomially where the major contribution comes from the first few terms...It follows that group velocity if proton is much less than that of the electron...

Please tell me if I am correct and sggest any other possible ways...

Regards,
neelakash
 
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Don't use v.
Use E=T+M and p=\sqrt{T^2+2M}.
 
but how would you compae between the group velocities?

Don't use v.
Use E=T+M and p=\sqrt{T^2+2M}.

I hope your formula is pc=√[K(K+2mc²)] where K is the KE

But, p=γmv...so that you are to know γ if you want to know v
γ s are different for e and p...
 
neelakash said:
but how would you compare between the group velocities?



I hope your formula is pc=√[K(K+2mc²)] where K is the KE

But, p=γmv...so that you are to know γ if you want to know v
γ s are different for e and p...
Sorry, I should have had T=\sqrt{T^2+2MT}.
I use T for KE, which is more common, and relativistic units with c absent.
v_P=E/p, and v_g=dE/dp=p/E.
You don't need gamma or v, but they are gamma=E/M and v=p/E.
 
Buddy,what you are using seems not quite effective here...Remember we are to compare between group and phase velocities of an e and a p whose KE are the same...And you have not used the fact that their KE are the same...

I am referring to another method...It is no approximation..stands on sheer logic...

1 stands for e and 2 stands for p

K=(γ₁-1)m₁c²=(γ₂-1)m₂c²
Now,(γ₂-1)/(γ₁-1)=m₁/m₂

=>(γ₂-γ₁)/(γ₁+γ₂-2)=(m₁-m₂)/(m₁+m₂)<0
Also,(γ₁+γ₂)>2

=> γ₂<γ₁
From which you can deduce the relation between group and phase velocity...
 

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